Testing whether linear equations are causal: A free probability theory approach

We propose a method that infers whether linear relations between two high-dimensional variables X and Y are due to a causal influence from X to Y or from Y to X. The earlier proposed so-called Trace Method is extended to the regime where the dimension of the observed variables exceeds the sample size. Based on previous work, we postulate conditions that characterize a causal relation between X and Y. Moreover, we describe a statistical test and argue that both causal directions are typically rejected if there is a common cause. A full theoretical analysis is presented for the deterministic case but our approach seems to be valid for the noisy case, too, for which we additionally present an approach based on a sparsity constraint. The discussed method yields promising results for both simulated and real world data.